- Research Article
- Open Access
Convergence of Paths for Perturbed Maximal Monotone Mappings in Hilbert Spaces
© Yuan Qing et al. 2010
- Received: 16 July 2010
- Accepted: 20 December 2010
- Published: 5 January 2011
Let be a Hilbert space and a nonempty closed convex subset of . Let be a maximal monotone mapping and a bounded demicontinuous strong pseudocontraction. Let be the unique solution to the equation . Then is bounded if and only if converges strongly to a zero point of A as which is the unique solution in , where denotes the zero set of , to the following variational inequality , for all .
- Hilbert Space
- Banach Space
- Variational Inequality
- Convex Subset
- Monotone Mapping
Throughout this work, we always assume that is a real Hilbert space, whose inner product and norm are denoted by and , respectively. Let be a nonempty closed convex subset of and a nonlinear mapping. We use and to denote the domain and the range of the mapping . and denote strong and weak convergence, respectively.
Recall the following well-known definitions.
(2)The single-valued mapping is maximal if the graph of is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if for , for every implies .
It is well-known that if is demicontinuous, then is hemicontinuous, however, the converse, in general, may not be true. In reflexive Banach spaces, for monotone mappings defined on the whole Banach space, demicontinuity is equivalent to hemicontinuity.
To find zeroes of maximal monotone operators is the central and important topics in nonlinear functional analysis. We observe that is a zero of the monotone mapping if and only if it is a fixed point of the pseudocontractive mapping . Consequently, considerable research works, especially, for the past 40 years or more, have been devoted to the existence and convergence of zero points for monotone mappings or fixed points of pseudocontractions, see, for instance, [1–23].
In 1965, Browder  proved the existence result of fixed point for demicontinuous pseudocontractions in Hilbert spaces. To be more precise, he proved the following theorem.
In 1968, Browder  proved the existence results of zero points for maximal monotone mappings in reflexive Banach spaces. To be more precise, he proved the following theorem.
For the existence of continuous paths for continuous pseudocontractions in Banach spaces, Morales and Jung  proved the following theorem.
Let be a Banach space. Suppose that is a nonempty closed convex subset of and is a continuous pseudocontraction satisfying the weakly inward condition. Then for each , there exists a unique continuous path , , which satisfies the following equation .
In 2002, Lan and Wu  partially improved the result of Morales and Jung  from continuous pseudocontractions to demicontinuous pseudocontractions in the framework of Hilbert spaces. To be more precise, they proved the following theorem.
Let be a bounded closed convex set in . Assume that is a demicontinuous weakly inward pseudocontractive map. Then has a fixed point in . Moreover; for every , defined by converges to a fixed point of .
In this work, motivated by Browder , Lan and Wu , Morales and Jung , Song and Chen , and Zhou [22, 23], we consider the existence of convergence of paths for maximal monotone mappings in the framework of real Hilbert spaces.
This completes the proof.
By using Lemma 2.1 and Theorem B2, we can obtain the desired conclusion easily.
From Lemma 2.2, one can obtain the desired conclusion easily.
This completes the proof.
Let be a nonempty closed convex subset of a Hilbert space and a maximal monotone mapping. Then . If one defines by , for all , then is a nonexpansive mapping with and , where denotes the set of fixed points of .
This completes the proof.
Set . Let denote the Banach space of all bounded real value functions on with the supremum norm, a subspace of , and an element in , where denotes the dual space of . Denote by the value of at . If , for all , sometimes will be denoted by . When contains constants, a linear functional on is called a mean on if . We also know that if contains constants, then the following are equivalent.
To prove our main results, we also need the following lemma.
Lemma 2.5 (see [20, Lemma 4.5.4]).
Now, we are in a position to prove the main results of this work.
From Theorem 2.6, we can obtain the following interesting fixed point theorem. The composition of bounded, demicontinuous, and strong pseudocontractions with the metric projection has a unique fixed point. That is, .
The third author was supported by the National Natural Science Foundation of China (Grant no. 10771050).
- Browder FE: Fixed-point theorems for noncompact mappings in Hilbert space. Proceedings of the National Academy of Sciences of the United States of America 1965, 53: 1272–1276. 10.1073/pnas.53.6.1272MathSciNetView ArticleMATHGoogle Scholar
- Browder FE: Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces. Archive for Rational Mechanics and Analysis 1967, 24: 82–90.MathSciNetView ArticleMATHGoogle Scholar
- Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. In Nonlinear Functional Analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968). American Mathematical Society, Providence, RI, USA; 1976:1–308.Google Scholar
- Browder FE: Nonlinear maximal monotone operators in Banach space. Mathematische Annalen 1968, 175: 89–113. 10.1007/BF01418765MathSciNetView ArticleMATHGoogle Scholar
- Browder FE: Nonlinear monotone and accretive operators in Banach spaces. Proceedings of the National Academy of Sciences of the United States of America 1968, 61: 388–393. 10.1073/pnas.61.2.388MathSciNetView ArticleMATHGoogle Scholar
- Bruck, RE Jr.: A strongly convergent iterative solution of for a maximal monotone operator in Hilbert space. Journal of Mathematical Analysis and Applications 1974, 48: 114–126. 10.1016/0022-247X(74)90219-4MathSciNetView ArticleMATHGoogle Scholar
- Chen R, Lin P-K, Song Y: An approximation method for strictly pseudocontractive mappings. Nonlinear Analysis: Theory, Methods & Applications 2006,64(11):2527–2535. 10.1016/j.na.2005.08.031MathSciNetView ArticleMATHGoogle Scholar
- Chidume CE, Moore C: Fixed point iteration for pseudocontractive maps. Proceedings of the American Mathematical Society 1999,127(4):1163–1170. 10.1090/S0002-9939-99-05050-9MathSciNetView ArticleMATHGoogle Scholar
- Chidume CE, Osilike MO: Nonlinear accretive and pseudo-contractive operator equations in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 1998,31(7):779–789. 10.1016/S0362-546X(97)00439-2MathSciNetView ArticleMATHGoogle Scholar
- Chidume CE, Zegeye H: Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps. Proceedings of the American Mathematical Society 2004,132(3):831–840. 10.1090/S0002-9939-03-07101-6MathSciNetView ArticleMATHGoogle Scholar
- Deimling K: Nonlinear Functional Analysis. Springer, Berlin, Germany; 1985:xiv+450.View ArticleMATHGoogle Scholar
- Deimling K: Zeros of accretive operators. Manuscripta Mathematica 1974, 13: 365–374. 10.1007/BF01171148MathSciNetView ArticleMATHGoogle Scholar
- Kato T: Demicontinuity, hemicontinuity and monotonicity. Bulletin of the American Mathematical Society 1964, 70: 548–550. 10.1090/S0002-9904-1964-11194-0MathSciNetView ArticleMATHGoogle Scholar
- Lan KQ, Wu JH: Convergence of approximants for demicontinuous pseudo-contractive maps in Hilbert spaces. Nonlinear Analysis: Theory, Methods & Applications 2002,49(6):737–746. 10.1016/S0362-546X(01)00130-4MathSciNetView ArticleMATHGoogle Scholar
- Morales CH, Jung JS: Convergence of paths for pseudocontractive mappings in Banach spaces. Proceedings of the American Mathematical Society 2000,128(11):3411–3419. 10.1090/S0002-9939-00-05573-8MathSciNetView ArticleMATHGoogle Scholar
- Morales CH, Chidume CE: Convergence of the steepest descent method for accretive operators. Proceedings of the American Mathematical Society 1999,127(12):3677–3683. 10.1090/S0002-9939-99-04975-8MathSciNetView ArticleMATHGoogle Scholar
- Qin X, Su Y: Approximation of a zero point of accretive operator in Banach spaces. Journal of Mathematical Analysis and Applications 2007,329(1):415–424. 10.1016/j.jmaa.2006.06.067MathSciNetView ArticleMATHGoogle Scholar
- Qin X, Kang SM, Cho YJ: Approximating zeros of monotone operators by proximal point algorithms. Journal of Global Optimization 2010,46(1):75–87. 10.1007/s10898-009-9410-6MathSciNetView ArticleMATHGoogle Scholar
- Song Y, Chen R: Convergence theorems of iterative algorithms for continuous pseudocontractive mappings. Nonlinear Analysis: Theory, Methods & Applications 2007,67(2):486–497. 10.1016/j.na.2006.06.009MathSciNetView ArticleMATHGoogle Scholar
- Takahashi W: Nonlinear Functional Analysis, Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar
- Zhou H: Iterative solutions of nonlinear equations involving strongly accretive operators without the Lipschitz assumption. Journal of Mathematical Analysis and Applications 1997,213(1):296–307. 10.1006/jmaa.1997.5539MathSciNetView ArticleMATHGoogle Scholar
- Zhou H: Convergence theorems of common fixed points for a finite family of Lipschitz pseudocontractions in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2008,68(10):2977–2983. 10.1016/j.na.2007.02.041MathSciNetView ArticleMATHGoogle Scholar
- Zhou H: Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces. Journal of Mathematical Analysis and Applications 2008,343(1):546–556. 10.1016/j.jmaa.2008.01.045MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.